Impulse and time integration
Impulse is the accumulated effect of force over time. A short large collision force and a longer smaller force can produce the same momentum change.
Think like a programmer
deltaTime to momentum, so fixed and well-tested time steps matter to collision and contact code.Model checklist
- Inputs
- Force samples in newtons and time intervals in seconds.
- State
- Current momentum.
- Rule
- Add force times time interval.
- Output
- Momentum change in kg·m/s.
- Check
- Constant force gives force × total time.
Elastic collision check
Change the incoming body. The collision model calculates outgoing velocities and checks total momentum before and after.
Momentum before 8.00 kg·m/s; after 8.00 kg·m/s. Outgoing velocities: 1.33 and 5.33 m/s.
Integrate a force ledger
At each sampled interval, record the force convention, start time, end time, and the body or system that receives the impulse. The discrete update is:
\[\mathbf p_{n+1}=\mathbf p_n+\mathbf F_n\Delta t, \qquad \mathbf J\approx\sum_n\mathbf F_n\Delta t_n.\]For a constant force, summing many intervals must equal force times total duration. For variable force, use the declared sample convention: left endpoint, midpoint, or a force measured over the whole interval. That choice is part of the model, particularly for a sharp collision pulse.
An impulse belongs to a boundary. Two colliding bodies receive equal and opposite internal impulses, so their combined momentum is unchanged in an isolated model. A wall exerts an external impulse on the selected two-body system; it becomes internal only when the wall or Earth is included. Log both the per-body momentum changes and the net system residual instead of relying on an animation’s apparent bounce.
Resolve contact deliberately
A direct restitution rule maps pre-contact velocities to post-contact velocities and is useful when only before/after state matters. A time-resolved contact model instead needs overlap or gap, normal direction, relative velocity, force law, damping, and a contact termination condition. Its peak force and contact duration depend on the chosen compliance and time step.
Test a constant-force pulse, equal/opposite isolated pair impulses, zero-duration/zero-force identity, and a wall event with the correct external ledger. Repeat a fixed-duration force history with smaller intervals and compare total impulse and final momentum. Do not infer material stress, sound, or deformation from an impulse-only model.
Try this experiment
Prediction: A larger incoming speed needs a larger momentum change to stop.
Increase incoming velocity. Compare the change implied by outgoing velocities, then explain which extra data a force-time simulation would need.Where this model breaks
Summary
Integrate a declared force history into momentum, keep system boundaries in the impulse ledger, and compare fixed-duration refinements. Use a direct collision rule only when its before/after assumptions answer the question.
Glossary
- Impulse: force accumulated over time.
- Momentum change: final momentum minus initial momentum.
- Contact duration: time two bodies exert collision forces.
Self-check
- What does force × time change?
- What constant-force case tests the accumulator?
- What detail does a before/after collision omit?
Sources
- OpenStax, University Physics Volume 1, impulse chapter.
Model contract
Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.
\[\text{observable output} = f(\text{inputs},\,\text{state})\]- Inputs
- Quantities you set or measure, with units and useful bounds.
- State
- Values the program must retain to reproduce the next result.
- Rule
- The relationship or update that turns inputs and state into a result.
- Check
- A known limit, unit check, invariant, or measured result that can expose a bad model.
Implement the idea as a model
For Impulse and Time Integration, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Momentum and Collisions, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.
Guided experiment
Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.
Where this model breaks
This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.
Summary
Treat Impulse and Time Integration as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.
Glossary
- Input: a measured value or chosen parameter supplied to a model.
- State: the smallest set of values needed to continue or reproduce a model.
- Validation: comparing an output with a known result, limit, invariant, or measurement.
Self-check
- Which values are inputs, and which values must remain state?
- What observable result would tell you the model is behaving as expected?
- Which assumption would you test first before applying the model to a real system?
Model review: turn Impulse and Time Integration into a test
Connect force over time to momentum change and model a short collision as an accumulated impulse.
- Name the inputs and units that the momentum and collisions model needs.
- Separate the state you must keep from values you can calculate when needed.
- Write one rule that maps the current state and inputs to an observable result.
- Choose a limiting case, unit check, invariant, or known result before trusting an output.
- State one assumption you would change before using this simplified model for a real decision.